The percentage error in measurements of length and time period is `2%` and `1%` respectively . The percentage error in measurements of 'g' is

Options

A. `2%`
B. `4%`
C. `6%`
D. `8%`

Answer

To find the percentage error in the measurement of 'g' given the percentage errors in length and time period, we can follow these steps:

### Step 1: Understand the relationship between the variables
The formula for gravitational acceleration 'g' in terms of length 'L' and time period 'T' is derived from the equation of a simple pendulum:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
Squaring both sides gives:
\[ T^2 = 4\pi^2 \frac{L}{g} \]
Rearranging this, we find:
\[ g = \frac{4\pi^2 L}{T^2} \]

### Step 2: Identify the errors in measurements
We are given:
- Percentage error in length (L) = 2%
- Percentage error in time period (T) = 1%

### Step 3: Use the formula for error propagation
To find the percentage error in 'g', we can use the formula for error propagation. The formula for the percentage error in 'g' can be derived as follows:
\[ \frac{\Delta g}{g} \times 100 = \frac{\Delta L}{L} \times 100 + 2 \times \frac{\Delta T}{T} \times 100 \]

### Step 4: Substitute the known values
Substituting the known percentage errors into the equation:
- Percentage error in L = 2%
- Percentage error in T = 1%

Thus,
\[ \text{Percentage error in } g = 2 + 2 \times 1 \]
\[ = 2 + 2 = 4\% \]

### Step 5: Conclusion
The percentage error in the measurement of 'g' is **4%**.

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